Do cabs run 24 hours a day in London?

We have had our hotel in Glasgow order us a cab for that time of the morning so probably the English can do the same thing.

You mean a cab from your hotel to the train station? You should be able to pre-arrange a cab ride to pick you up and pre-pay with a CC. That's what my daughter and I did when we were in London in 2012. Just don't forget your passport! (we did) Then it was a huge hassel to get our cab to turn around so we could retrieve it. But you can't leave the country without the passport- even for a day trip. ;-)

I don't know how tightly they enforce the cutoff time for the first 0540 train but Eurostar are well known for being very strict on their minimum 30 minute cutoff for check in. There is plenty to do after you check in, passport control, security including various scans, and getting to the right area.

Food is available after security as are coffees and water; you aren't allowed to take open drinks through security.

Get there with plenty of time to spare.

Cabs and minicabs run all night, with appropriate adjustments to the fare.

Yes it is. Pre arrange with hotel or make a mental note where you can flag down a cabbie near where ever you've chosen to stay. You will need to be at the train station a minimum of 30 minutes early so plan accordingly.